RTG Seminar Lecture Notes: The Positive Mass Theorem of General Relativity

General Relativity is the study of Lorentz 4-manifolds (M, g) whose metric arises from a solution to the Einstein equations. We will take the signature of the metric to be (− + ++), where the minus sign indicates the time component. Then the tangent space at any point may be divided into three different types of vectors, namely timelike, spacelike, and null vectors whose lengths squared respectively satisfy g(v, v) < 0 (> 0) (= 0). By assumption matter particles travel on timelike geodesics, light and radiation travel on null geodesics, and nothing travels faster than the speed of light. The source for the gravitational field is a symmetric 2-tensor T , referred to as the stress-energy-momentum tensor. It is physically defined in the following way. If v, w ∈ TpM are future directed unit timelike vectors, then T (v, w) represents the energy-momentum density in the direction v as measured by an observer moving in the direction w . In order to have a viable physical theory two natural restrictions are usually placed on the stress tensor. The first asserts that